The Precise Deﬁnition of Limit: δ-Proofs Math 109
Dr. Evans Spring 2007
The precise deﬁnition of limit given on page 50 in section 1.8 was developed over hundreds of years. It is therefore no surprise that students often have diﬃculties quickly mastering the deﬁnition and how to use it. The following remarks and examples are meant to serve as a guide as you learn to use use the deﬁnition towrite your own δ-proofs. Please be aware as you study this that styles in proof writing vary greatly, and no two people will ever write the exact same proof. However, the logical structure of any two proofs (of the same result) should be more or less the same. In the present setting, you must demonstrate, usually through a series of algebraic steps, that the implication in the deﬁnition holds. Inan δ-proof, you must ﬁrst do some calculations to ﬁnd the number δ, but these calculations are not part of the proof. Instead, the proof consists of specifying a value for δ in terms of and showing that the implication in the limit deﬁnition holds for this value of δ. Let’s begin by making a few remarks about absolute value. Absolute Value. First recall that if x is a real number, the absolutevalue of x is the distance from x to 0 and is written |x|. Said another way, we can deﬁne |x| = Therefore, if c is any real number, we have |x − c| = x − c if x ≥ c; c − x if x < c x if x ≥ 0; −x if x < 0.
so that it is natural (and useful) to think of |x − c| as the distance from x to c. Two important equivalences involving absolute value are |x − c| < δ ⇐⇒ −δ < x − c < δ ⇐⇒ c − δ < x < c + δwhere the symbol ⇐⇒ means “is equivalent to”. In words, these equivalences say that x is less than δ units from c if and only if the diﬀerence x − c is between −δ and δ if and only if x is in the interval (c − δ, c + δ). Draw a picture! The Deﬁnition. Let us state the deﬁnition of limit, ﬁrst informally and then precisely. Deﬁnition (informal). If f (x) is a function deﬁned for all values of x nearx = c, except perhaps at x = c, and if L is a real number such that the values of f (x) get closer and closer to L as the values of x are taken closer and closer to c, then we say L is the limit of f (x) as x approaches c and we write lim f (x) = L.
To transform this intuitive idea into a precise deﬁnition, we need to say exactly what we mean by “ f (x) gets closer and closer to L as thevalues of x are taken closer and closer to c”. The main idea is to notice that if two quantities are getting “closer and closer”, then the distance between them is becoming “smaller and smaller”. That is, the distance is eventually smaller than any speciﬁed positive number. Note that there is an implication in this informal deﬁnition. Namely it says if we allow x to become closer and closer to c,then f (x) will become closer and closer to L. When we write a proof, we show that by taking x suﬃciently close to c, we make f (x) arbitrarily close to L. However, before we can demonstrate the implication in the
The Precise Deﬁnition of Limit
deﬁnition, we need to know how close to c is suﬃciently close; that is we need to ﬁnd δ. Now let us state the precise deﬁnition.Deﬁnition. Suppose that c and L are real numbers and f (x) is a function deﬁned in an open interval containing c, except perhaps at x = c. If for every positive number > 0, there exists a positive number δ > 0 (which depends on ) such that 0 < |x − c| < δ implies |f (x) − L| < ,
then we say that L is the limit of f (x) as x approaches c and we write lim f (x) = L.
Examples. Now we will write afew proofs to guide you in your own writing. To emphasize the logical structure of the proof, we will not show how we found our δ in the ﬁrst two examples. Example 1. Show that lim (3x − 5) = 1.
> 0 and deﬁne δ = /3. Then if 0 < |x − 2| < δ, we have |(3x − 5) − 1| = |3x − 6| = 3|x − 2| < 3( /3) = . (since |x − 2| < δ and δ = /3)
Therefore we have shown 0 < |x − 2| < δ...
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